10. Odd, Even, Pos, Neg Flashcards
What are integers?
Positive (1, 2, 3, etc.) or negative (-1, -2, -3, etc.) whole numbers that have no fractional part. Integers include counting numbers, their negative counterparts and zero
- Adding, subtracting and multiplying integers always give you an integer
- Dividing an integer by another integer results in an integer or a fraction
What is an even number?
Integers that are divisible by 2. 14 is even because 14/2 = integer (7)
What is an odd number?
Integers that are not divisible by 2. 15 is not even because 15/2 does not equal an integer (7.5)
What are the addition/subtraction rules for odds and evens?
- Add or subtract 2 odds or 2 evens, and the result is EVEN
- Add or subtract an odd with an even, and the result is ODD
Even ± Even = Even (e.g. 2 + 2 = 4)
Odd ± Odd = Even (e.g. 3 + 3 = 6
Even ± Odd = Odd (e.g. 2 + 3 = 5)
What are the multiplication rules for odds and evens?
- When you multiply integers, if ANY of the integers is even, the result is EVEN
- Likewise, if NONE of the integers is even, then the result is ODD
Even * Even = Even (e.g. 2 * 2 = 4) ALSO divisible by 4
Even * Odd = Even (e.g. 2 * 3 = 6)
Odd * Odd = Odd (e.g. 3 * 3 = 9)
*NOTE: if you multiply together several even integers, the result will be divisible by higher and higher powers of 2 (e.g. 2 * 5 * 6 * 10 = 600 which is divisible by 8 = 2^3)
What are the division rules for odds and evens?
Even / Even = Anything (even, odd or not an integer)
Even / Odd = Even or not an integer
Odd / Even = Not an integer
Odd / Odd = Odd or not an integer
*NOTE: an odd integer divided by any other integer CANNOT produce an even integer
What do you know about (prime + prime) = odd?
- One of those primes must be the number 2
- Conversely, if you know that 2 CANNOT be one of the primes in the sum, then the sum of the two primes must be even
If a and b are both prime numbers greater than 10, which of the following CANNOT be true?
I. ab is an even number
II. the difference between a and b = 117
III. the sum of a and b is even
I and II Only. a and b must be odd
I. ab must be an odd number (CANNOT be true)
II. odd – odd = even (CANNOT be true)
III. odd + odd = even (TRUE)
If x > 1, what is the value of integer x?
(1) There are x unique factors of x
(2) The sum of x and any prime number larger than x is odd
Answer A.
(1) SUFFICIENT. In order for this to be true, every integer between 1 and x, inclusive, must be a factor of x. Testing numbers you can see that this property holds for 1 and for 2 only. Therefore, this x = 1 or 2. However, the original problem says that x > 1, so x must equal 2
(2) INSUFFICIENT. x must equal at least 2, so this includes prime numbers larger than 2. Therefore, the prime number is odd, and x is even. However, this does not tell you which even number x could be
What is the remainder when a is divided by 4?
(1) a is the square of an odd integer
(2) a is a multiple of 3
Answer A. Even numbers are multiples of 2, so any arbitrary even number can be written as 2n, where n is any integer. Odd numbers are one more or less than multiples of 2, so an arbitrary odd number can be written as 2n + 1 or 2n – 1, where n is an integer
(1) SUFFICIENT. The square of an arbitrarily odd number can be written as (2n + 1)^2 = 4n^2 + 4n + 1. The first two terms of this expression are multiples of 4, which have a remainder of 0 when divided by 4. The third term, 1, gives a remainder of 1 when divided by 4.
(2) INSUFFICIENT. Test numbers. When 3 is divided by 4, the remainder is 3. When 6 is divided by 4, the remainder is 2.
If a, b, and c are integers and ab + c is odd, which of the following must be true?
I. a + c is odd
II. b + c is odd
III. abc is even
III Only. A table is a good tool for keeping track of different scenarios. Two steps: (1) ask yourself, what would need to be true in order for ab + c to be odd? An odd plus and even equals an odd, so either ab or c needs to be even (2) write out only those scenarios in which either ab or c is even
(1) a = ODD, b = ODD, c = EVEN, ab + c = ODD
(2) a = ODD, b = EVEN, c = ODD, ab + c = ODD
(3) a = EVEN, b = ODD, c = ODD, ab + c = ODD
(4) a = EVEN, b = EVEN, c = ODD, ab + c = ODD
I. a + c is odd: Scenario (2) goes against this statement
II. b + c id odd: Scenario (3) goes against this statement
III. abc is even: For abc to be even, only one of the three variables needs to be even. In all four scenarios, at least one of the variables is even. MUST BE TRUE
How should you deal with the scenarios in an odds and evens problem?
(1) Ask yourself: how do any given constraints limit the possible scenarios?
(2) Create a table to keep track of the allowable scenarios, given any constraints in the problem
Number Properties Guide, Ch 2, Q 9. If x, y, and z are prime numbers and x < y < z, what is the value of x?
(1) xy is even
(2) xz is even
Answer D.
(1) SUFFICIENT. If xy is even, then x is even or y is even. Since x < y, x must be 2, because 2 is the smallest and only even prime number
(2) SUFFICIENT. Similarly, if xz is even, then x is even or z is even. Since x < z, x must equal 2, because 2 is the smallest and only even prime number
Number Properties Guide, Ch 5, Q 1. If x, y and z are integers, is x even?
(1) 10^x = (4^y)(5^z)
(2) 3^(x + 5) = 27^(y + 1)
Answer A. Odds & Evens.
(1) SUFFICIENT. You can break the bases down into prime factors: (2^x)(5^x) = (2^2y)(5^z). This means that x = 2y and x = z. y is an integer, so x must be even, because x = 2y
(2) INSUFFICIENT. You can again break the bases down into prime factors. 3^(x + 5) = 3^(3y + 3). This tells you that x + 5 = 3y + 3, so x + 2 = 3y. y is an integer, so x must be 2 larger than a multiple of 3, but that does not tell you whether x is even. If y = 1, then x = 1 (odd), but if y = 2, then x = 4 (even)
Number Properties Guide, Ch 5, Q 3. If c and d are integers, is c – 3d even?
(1) c and d are odd
(2) c – 2d is odd
Answer A. Odds & Evens.
(1) SUFFICIENT. If both c and d are odd, then c – 3d = Odd – (3 * Odd) = Odd – Odd = Even
(2) INSUFFICIENT. If c – 2d is odd, then c must be odd, because 2d will always be even. However, this tells you nothing about d